University of Florida/Egm4313/s12.team11.R5

Report 5

Intermediate Engineering Analysis
Section 7566
Team 11
Due date: March 30, 2012.

R5.1

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Problem Statement

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Given: Find   for the following series:

1.  
2.  

Find   for the Taylor series of

3.   at  

4.   at  

5.   at  

Solution

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The radius of convergence   is defined as

 

1.  
 
 
 
 

                                                           

2.  
 
 
 
 

However, in this problem, the series   term is   not  , as is the general form.
Therefore, this implies:

 
 

                                                         


3. The Taylor series for   is expressed as  

 

 

 

 

Therefore:  

                                                            

4. The Taylor series for   at   is expressed as  

 

 

 

 

 

                                                              



5. The Taylor series for   at   is expressed as  

 

 

 

 

 

For convergence:  

 

 

Therefore,

                                                              



R5.2

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Solved by: Andrea Vargas

Problem Statement

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Part 1:Determine whether the following are linearly independent using the Wronskian

Part 2: Determine whether the following are linearly independent using the Gramian

Solution

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Part 1

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Using the Wronskian we check for linear independence.

We know from (1) and (2) in 7-35 that if
 
Then the functions are linearly independent.

Part 1.1
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Taking the derivatives of each function:

 
 

 

                                            f(x) and g(x) are linearly independent
Part 1.2
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Taking the derivatives of each function:

 
 

 

                                             f(x) and g(x) are linearly independent

Part 2

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Using the Gramian we check for linear independence.
We know from the notes in (1) 7-34 that:
 

and that the Gramian is defined as:  

Then f,g are linearly independent if  

Part 2.1
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Taking scalar products:
 
 
 

 

                                            f(x) and g(x) are linearly independent
Part 2.2
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Taking scalar products:
 
We can use the trig identity for power reduction  
Then we have,
 
 

 
 
 

 
 
 

 

                                            f(x) and g(x) are linearly independent

Conclusion

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By both methods (the Wronskian and the Gramian) we obtain the same results.

R5.3

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Problem Statement

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Verify using the Gramian that the following two vectors are linearly independent.

 
 

Solution

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We know from (3) 8-9 that:
 

We obtain,
 
   

Then,
 

                                             b_1 and b_2 are linearly independent

R5.4

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Problem Statement

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Show that   is indeed the overall particular solution of the L2-ODE-VC   with the excitation  .

Discuss the choice of   in the above table e.g., for   why would you need to have both   and   in   ?

Solution

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Because the ODE is a linear equation in y and its derivatives with respect to x, the superposition principle can be applied:

  is a specific excitation with known form of   and   is a specific excitation with known form of  

 

becomes

 

proving that

  is indeed the overall particular solution of the L2-ODE-VC   with the excitation  

According to Fourier Theorem periodic functions can be represented as infinite series in terms of cosines and sines:

 

where the coefficients   are the Fourier coefficients calculated using Euler formulas.

So even though the system is being excited by functions like   the particular solution would still include both   and   in   because the excitation is a periodic function that can be represented as the Fourier infinite series in terms of both   and   times the Fourier coefficients

R5.5

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Part 1

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Problem Statement

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Show that   and   are linearly independant using the Wronskian and the Gramain (integrate over 1 period)

Solution

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One period of  
Wronskian of f and g
 

Plugging in values for  
   
 
 

 They are linearly Independant using the Wronskian.

 
 
 
 
 
 
 

 They are linearly Independent using the Gramain.

Problem Statement

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Find 2 equations for the 2 unknowns M,N and solve for M,N.

Solution

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Plugging these values into the equation given ( ) yields;
 
Simplifying and the equating the coefficients relating sin and cos results in;
 
 
Solving for M and N results in;

   

Problem Statement

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Find the overall solution   that corresponds to the initial conditions  . Plot over three periods.

Solution

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From before, one period   so therefore, three periods is  
Using the roots given in the notes  , the homogenous solution becomes;
 
Using initial condtion  ;
 
 
with  
 
Solving for the constants;
 
 
Using the   found in the last part;
 

  

 

 

R5.6

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solved by Luca Imponenti

Problem Statement

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Complete the solution to the following problem

 

where

 

and

 

Find the overall solution   corresponds to the initial condition:

 

Plot the solution over 3 periods.

Particular Solution

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Taking the derivatives of the particular solution  

 

 

 

Plugging these into the ODE yields

     

Equating like terms allows us to solve for M and N

 

 

 

 

 

So the particular solution is

 

Overall Solution

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The overall solution in the sum of the homogeneous and particular solutions

 

 

To find A and B we apply the initial conditions

 

 

Taking the derivative

 

 

 

 

Giving us the overall solution

    

Plot

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The period for   is  

Plotting the solution   over 3 periods yields

 

R5.7

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Solved by Daniel Suh

Problem Statement

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1. Find the components   using the Gram matrix.
2. Verify the result by using   and  , and rely on the non-zero determinant matrix of   and   relative to the bases of   and  .

Part 1 Solution

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Gram Matrix

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Thus,

 

 

 

 

 

Defining c

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Define:  
 

If  , then   exists

 

Finding c

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  thus,  exists

 

 

                                                      

Part 2 Solution

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                                                       solution is correct

R5.8

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Problem Statement

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Find the integral

  for   and  

Using integration by parts, and then with the help of of

General Binomial Theorem

 

Solution

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For  :

 

For substitution by parts,  

 

 

 

Therefore:

                                      

Using the General Binomial Theorem:

 

Therefore:  

Which we have previously found that answer as:

                                      




For  :

 

Initially we use the following substitutions:  

 

First let us consider the first term:  

Next, we use the integration by parts:  
 

 

 

Next let us consider the second term:  

Again, we will use integration by parts:  
 

 

 

Therefore:

 

 

Re-substituting for t:

 

 

 

Therefore:

                                      



Using the General Binomial Theorem for the integral with t substitution  :

 

Therefore:  

Which we have previously found that answer as:

                                      

R5.9

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Solved by: Gonzalo Perez

Problem Statement

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Consider the L2-ODE-CC (5) p.7b-7 with   as excitation:

  (5) p.7b-7

  (1) p.7c-28

and the initial conditions

 .

Part 1

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Part A

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Project the excitation   on the polynomial basis

  (1)

i.e., find   such that

  (2)

for x in  , and for n = 3, 6, 9.

Plot   and   to show uniform approximation and convergence.

Note that:

  (3)

Solution

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To solve this problem, it is important to know that the scalar product is defined as the following:

 .

Therefore, it follows that:

 , where   and  .

We know that if   are linearly independent, then by theorem on p.7c-37, the matrix is solvable.

According to this and (3)p.8-14:

If   exists  . (3)p.8-14

Now let's define the Gram matrix   as a function of  :

  (1)p.8-13

Defining the "d" matrix as was done in (3)p.8-13, we get:

 . (3)p.8-13

And according to (1)p.8-15:   (1)p.8-15

Now, we can find the values to compare   to  .

Using Matlab, this is the code that was used to produce the results:

The Matlab code above produced the following graph:

Where   is represented by the dashed line and the approximation, , is represented by the red line. This code can work for all n values.

Part B

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In a seperate series of plots, compare the approximation of the function   by Taylor series expansion about  .

Where:  

Solution
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For n=1:

 

For n=2:

 

For n=3:

 

For n=4:

 

For n=5:

 

For n=6:

 

For n=7:

 

For n=8:

 

For n=9:

 

For n=10:

 

For n=11:

 

For n=12:

   

For n=13:

   

For n=14:

   

For n=15:

   

For n=16:

   

Using Matlab to plot the graph:

Part 2

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Find   such that:

  (1) p.7c-27

with the same initial conditions as in (2) p.7c-28.

Plot   for n = 3, 6, 9, for x in  .

In a series of separate plots, compare the results obtained with the projected excitation on polynomial basis to those with truncated Taylor series of the excitation. Plot also the numerical solution as a baseline for comparison.

Solution

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First, we find the homogeneous solution to the ODE:
The characteristic equation is:
 
 
Then,  
Therefore the homogeneous solution is:
 

Now to find the particulate solution

For n=3:

 

 

We can then use a matrix to organize the known coefficients:

 

Then, using MATLAB and the backlash operator we can solve for these unknowns:
 
Therefore
 

Superposing the homogeneous and particulate solution we get
 

Differentiating:
  Evaluating at the initial conditions:
 
 

We obtain:
 
 

Finally we have:
 

For n=6:

 

 

We can then use a matrix to organize the known coefficients:

 

Then, using MATLAB and the backlash operator we can solve for these unknowns:
 
Therefore
 

Superposing the homogeneous and particulate solution we get
 

Differentiating:
  Evaluating at the initial conditions:
 
 

We obtain:
 
 

Finally
 

For n=9:

 

 

We can then use a matrix to organize the known coefficients:
   

Then, using MATLAB and the backlash operator we can solve for these unknowns:
 
Therefore
 
 

Superposing the homogeneous and particulate solution we get
 
 

Differentiating:
   
Evaluating at the initial conditions:
 
 

We obtain:
 
 

Finally
 
 

Here is the graph for this problem using Matlab: